We develop a set of new motion estimation (ME) algorithms based mainly on Unsymmetric-Cross Multi-Hexagon-Grid Search (UMH). The original algorithms are improved by applying the Successive Elimination Algorithm (SEA) and subsampling the image blocks while computing the matching criterion. We also improve SEA by adding a small constant to the lower bound before trying to eliminate the current checking point. The motion compensated results stay in most cases similar to the original UMH algorithm, while computation is decreased by up to 95%. The new algorithms outperform in both image quality and computational efficiency other well-known fast ME algorithms such as Three Step Search (TSS), Diamond Search (DS), and Hexagon-Based Search (HEXBS).